| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two-sample z-test (known variances) |
| Difficulty | Moderate -0.3 This is a straightforward application of standard results for sampling distributions and linear combinations of normal variables. Part (i) requires stating that sample means are normally distributed with known parameters (direct recall). Part (ii) involves finding the distribution of a difference of sample means and calculating a probability using tables—routine S3 content with no problem-solving insight required, though the multi-step calculation and handling of two samples makes it slightly more involved than basic single-variable questions. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\bar{X}_L \sqcap N(5, \frac{0.7^2}{20})\) | B1 | If no parameters allow in (ii) |
| \(\bar{X}_E \sqcap N(4.5, \frac{0.5^2}{25})\) | B1 | If 0.7/20, 0.5/25 then B1 for both, with means in (ii) |
| 2 marks total | ||
| (ii) Use \(\bar{X}_L - \bar{X}_E \sqcap N(0.5, \sigma^2)\) | M1A1 | QR: \(\bar{X}_L - \bar{X}_E \sqcap N(0.5, \sigma^2)\) cao |
| \(\sigma^2 = 0.49/20 + 0.25/25\) | B1 | |
| \(1 - \Phi([1-0.5]/\sigma) = 0.0036\) or \(0.0035\) | M1 | RH probability implied. If 0.7, 0.5 in \(\sigma^2\), M1A1B0M1A1 for 0.165 |
| 5 marks total |
(i) $\bar{X}_L \sqcap N(5, \frac{0.7^2}{20})$ | B1 | If no parameters allow in (ii)
$\bar{X}_E \sqcap N(4.5, \frac{0.5^2}{25})$ | B1 | If 0.7/20, 0.5/25 then B1 for both, with means in (ii)
| | 2 marks total |
(ii) Use $\bar{X}_L - \bar{X}_E \sqcap N(0.5, \sigma^2)$ | M1A1 | QR: $\bar{X}_L - \bar{X}_E \sqcap N(0.5, \sigma^2)$ cao
$\sigma^2 = 0.49/20 + 0.25/25$ | B1 |
$1 - \Phi([1-0.5]/\sigma) = 0.0036$ or $0.0035$ | M1 | RH probability implied. If 0.7, 0.5 in $\sigma^2$, M1A1B0M1A1 for 0.165
| | 5 marks total |2 Two brands of car battery, 'Invincible' and 'Excelsior', have lifetimes which are normally distributed. Invincible batteries have a mean lifetime of 5 years with standard deviation 0.7 years. Excelsior batteries have a mean lifetime of 4.5 years with standard deviation 0.5 years. Random samples of 20 Invincible batteries and 25 Excelsior batteries are selected and the sample mean lifetimes are $\bar { X } _ { I }$ years and $\bar { X } _ { E }$ years respectively.\\
(i) State the distributions of $\bar { X } _ { I }$ and $\bar { X } _ { E }$.\\
(ii) Calculate $\mathrm { P } \left( \bar { X } _ { I } - \bar { X } _ { E } \geqslant 1 \right)$.
\hfill \mbox{\textit{OCR S3 2007 Q2 [7]}}